PSI - Issue 12

M. Catena et al. / Procedia Structural Integrity 12 (2018) 538–552

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M. Catena et Al. / Structural Integrity Procedia 00 (2018) 000–000

Figure 3. Forces subvector components and unit cell deformed shapes of bending moment (a) and shear forces (b) transmission modes.

The principal vector s b pertaining to the pure bending mode has displacement and force sub-vectors respectively given by: d b = 1 / 2 [ 0 , α, 0 , α, 0 ] T , f b = α [ 0 , β r / 2 , 0 , 2 η r , 0 ] T (1) where α is the rigid unit cell rotation, β r = E r A r l 2 s / l r and η r = E r I r / l r . Eq. (1) indicates that due to bending, the top and bottom nodes of the unit cell rotate exactly of the same angle as the cross section they belong to does. In other words, the cell transfers the bending moments without deformations of the webs (Fig. 3). This result was already observed in Gesualdo et al. (2017) by numerical experimentation on Vierendeel girders unit cells. In addition, two bending moments are transferred through the unit cell: the first one, to which we refer as primary bending moment, is generated by the couple of axial forces acting on the rails, the second one, to which instead we refer as secondary or micro-polar moment, is exclusively due to curvature change of the rails and is given by the resultant of the nodal moments. The principal vector s V defining the shear transmission mode is generated by s b and has displacements and forces sub-vectors respectively given by d V = 0 , ∆ ψ + ϕ V , 0 , ϕ V , 0 T , f V = α 0 , − β r 2 , β r / 2 + 2 η r l r , − 2 η r , 0 T where The deformed shape of the unit cell due to the shear transmission mode is schematically depicted in Fig. 3(b) assuming that ∆ v i = v i + 1 − v i = 0. Actually, this shape is obtainable by superposition of two independent modes. The first one, shown in Fig. (4a), involves rigid rotation of the sleepers and skew bending deformation of both rails, whose end sections rotate exactly of the same angle as the sleepers do. Consequently, it generates respectively the transversal shear forces V = ± 24 η c l r ¯ ψ − ∆ v l r , with ¯ ψ rotation angle of the cell webs, and the secondary bending moments ∓ 12 η r ¯ ψ − ∆ v / l r applied on the left and right sides of the cell. The second deformed shape involved in shear transmission is instead associated with the relative rotation ξ = ( ψ − ϕ ) of the cell nodal sections with respect to the mean nodal rotation ϕ . It does not produce any transversal shear (Fig. 4b) and is characterized by rigid translations of the rails and flexural deformation only of the sleepers. Their end sections in this mode does not rotate but carry out a relative transversal displacement equal to ( ψ − ϕ ) l s . The related distorting forces acting on each cell face are a couple of axial forces N V = ± 12 ( η s / l s ) ( ψ − ϕ ) and two nodal moments 6 η s ( ψ − ϕ ) equilibrating the primary bending moment generated by the forces N V . It is noteworthy that, as consequence of the previous forces system, an uniform horizontal shear S H = N V is induced in each sleeper. ∆ ψ = + 8 with η s = ( E s I s ) / [ l s (6 α s − 1)] and α s = ( E s I s ) / ( k ϑ l s ) + 1 / 3. α 48 β r η t , ϕ V = α 48 β r η r